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Theorem gepsup 25353
Description: The greatest element of a poset is the supremum of the poset. (Contributed by FL, 19-Sep-2011.)
Hypothesis
Ref Expression
gepsup.1  |-  X  =  dom  R
Assertion
Ref Expression
gepsup  |-  ( R  e.  A  ->  ( ge `  R )  =  ( R  sup w  X ) )

Proof of Theorem gepsup
Dummy variable  r is distinct from all other variables.
StepHypRef Expression
1 elex 2809 . 2  |-  ( R  e.  A  ->  R  e.  _V )
2 id 19 . . . 4  |-  ( r  =  R  ->  r  =  R )
3 dmeq 4895 . . . . 5  |-  ( r  =  R  ->  dom  r  =  dom  R )
4 gepsup.1 . . . . 5  |-  X  =  dom  R
53, 4syl6eqr 2346 . . . 4  |-  ( r  =  R  ->  dom  r  =  X )
62, 5oveq12d 5892 . . 3  |-  ( r  =  R  ->  (
r  sup w  dom  r
)  =  ( R  sup w  X ) )
7 df-ge 25351 . . 3  |-  ge  =  ( r  e.  _V  |->  ( r  sup w  dom  r ) )
8 ovex 5899 . . 3  |-  ( R  sup w  X )  e.  _V
96, 7, 8fvmpt 5618 . 2  |-  ( R  e.  _V  ->  ( ge `  R )  =  ( R  sup w  X ) )
101, 9syl 15 1  |-  ( R  e.  A  ->  ( ge `  R )  =  ( R  sup w  X ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   _Vcvv 2801   dom cdm 4705   ` cfv 5271  (class class class)co 5874    sup w cspw 14319   gecge 25323
This theorem is referenced by:  sege  25355  defge3  25374
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-ge 25351
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